Câu hỏi của Cô Bé Yêu Đời

Question

Cho tổng $M=2+2^2+2^3+2^4+...+2^{100}$a) Tổng M có chia hết cho 31 không ? Vì sao ?b) Tính tổng M

Nguyễn Huy Tú · Accepted Answer

a) $M=2+2^2+2^3+...+2^{100}$$\Rightarrow M=\left(2+2^2+2^3+2^4+2^5\right)+...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)$$\Rightarrow M=2\left(1+2+2^2+2^3+2^4\right)+...+2^{96}\left(1+2+2^2+2^3+2^4\right)$$\Rightarrow M=2.31+...+2^{96}.31$$\Rightarrow M=\left(2+...+2^{96}\right).31⋮31$$\Rightarrow M⋮31$b) $M=2+2^2+2^3+...+2^{100}$$\Rightarrow2M=2^2+2^3+2^4+...+2^{101}$$\Rightarrow2M-M=\left(2^2+2^3+2^4+...+2^{101}\right)-\left(2+2^2+2^3+...+2^{100}\right)$$\Rightarrow M=2^{101}-2$Câu trả lời: a) $M=2+2^2+2^3+...+2^{100}$$\Rightarrow M=\left(2+2^2+2^3+2^4+2^5\right)+...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)$$\Rightarrow M=2\left(1+2+2^2+2^3+2^4\right)+...+2^{96}\left(1+2+2^2+2^3+2^4\right)$$\Rightarrow M=2.31+...+2^{96}.31$$\Rightarrow M=\left(2+...+2^{96}\right).31⋮31$$\Rightarrow M⋮31$b) $M=2+2^2+2^3+...+2^{100}$$\Rightarrow2M=2^2+2^3+2^4+...+2^{101}$$\Rightarrow2M-M=\left(2^2+2^3+2^4+...+2^{101}\right)-\left(2+2^2+2^3+...+2^{100}\right)$$\Rightarrow M=2^{101}-2$

Nguyễn Đình Dũng · Answer

a) M = 2 + 22 + 23 + ... + 2100= (2+22+23+24+25) + (26+27+28+29+210) + ... + (296+297+298+299+2100)= 2(1+2+22+23+24) + 26(1+2+22+23+24) + ... + 296(1+2+22+23+24)= 31(2+26+...+296) $⋮$ 31b) M = 2 + 22 + ... + 2100=> 2M = 22 + 23 + ... + 2101=> 2M - M = 2101 - 2=> M = 2101 - 2Câu trả lời: a) M = 2 + 22 + 23 + ... + 2100= (2+22+23+24+25) + (26+27+28+29+210) + ... + (296+297+298+299+2100)= 2(1+2+22+23+24) + 26(1+2+22+23+24) + ... + 296(1+2+22+23+24)= 31(2+26+...+296) $⋮$ 31b) M = 2 + 22 + ... + 2100=> 2M = 22 + 23 + ... + 2101=> 2M - M = 2101 - 2=> M = 2101 - 2

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